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arXiv:1907.11701 · European Journal of Physics

Solving the one-dimensional Ising chain via mathematical induction: an intuitive approach to the transfer matrix

Wenlong Wang, Rogelio Díaz-Méndez, Raudys R. Capdevila

Published Jul 17, 2019• 11 citations
arXivHTMLPDF

Abstract

Abstract The aim of this work is to present a formulation to solve the one-dimensional Ising model using the elementary technique of mathematical induction. This formulation is physically clear and leads to the same partition function form as the transfer matrix method, which is a common subject in the introductory courses of statistical mechanics. In this way our formulation is a useful tool to complement the traditional more abstract transfer matrix method. The method can be straightforwardly generalised to other short-range chains, coupled chains and is also computationally friendly. These two approaches provide a more complete understanding of the system, and therefore our work can be of broad interest for undergraduate teaching in statistical mechanics.

DOI
10.1088/1361-6404/ab330c

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Abstract The aim of this work is to present a formulation to solve the one-dimensional Ising model using the elementary technique of mathematical induction. This formulation is physically clear and leads to the same partition function form as the transfer matrix method, which is a common subject in the introductory courses of statistical mechanics. In this way our formulation is a useful tool to complement the traditional more abstract transfer matrix method. The method can be straightforwardly generalised to other short-range chains, coupled chains and is also computationally friendly. These two approaches provide a more complete understanding of the system, and therefore our work can be of broad interest for undergraduate teaching in statistical mechanics.
Abstract In this chapter we study the solution of the two-dimensional Ising model by means of the transfer matrix. Unlike the methods discussed in the previous chapter, the transfer matrix approach has greater generality and can used to solve exactly other two-dimensional models. Even if the general ideas behind this approach have been explained in Chapter 2 by means of the one-dimensional case, their application to the two-dimensional cases requires more powerful and sophisticated mathematical tools: for instance, the study the eigenvalues of the transfer matrix in the Ising model for T = Tcneeds to employ elliptic functions. The same is also true for other models. In order to present in the simplest possible way the main lines of this method, in the following we focus attention only on the solution of the model at T = Tcbecause this case can be analyzed in terms of simple trigonometric functions.
Abstract The Ising model is a well known statistical model which can be solved exactly by various methods. The most familiar one is the transfer matrix method. Sometimes it can be difficult to approach the open boundary case rather than periodic boundary ones in higher dimensions. But physically it is more intuitive to study the open boundary case, as it gives a closer view of the real system. We have introduced a new method called the pairing method to determine the exact partition function for the simplest case, a 1D Ising lattice. This method simplifies the problem's complexities and reduces it to a pure combinatorial problem. The study also reveals that it is possible to apply this pairing method in the case of a 2D square lattice. The obtained results agree perfectly with the values in the literature and this new approach provides an algorithmic insight to deal with such problems.
Abstract In undergraduate statistical mechanics courses the Ising model always plays an important role because it is the simplest non-trivial model used to describe magnetic systems. The one-dimensional model is easily solved analytically, while the two-dimensional one can be solved exactly by the Onsager solution. For this reason, numerical simulations are usually used to solve the two-dimensional model. Keeping in mind that the two-dimensional model is the platform for studying phase transitions, it is usually an exercise in computational undergraduate courses because its numerical solution is relatively simple to implement and its critical exponents are perfectly known. The purpose of this article is to present a detailed numerical study of the second-order phase transition in the two-dimensional Ising model at an undergraduate level, allowing readers not only to compare the mean-field solution, the exact solution and the numerical one through a complete study of the order parameter, the correlation function and finite-size scaling, but to present the techniques, along with hints and tips, for solving it themselves. We present the elementary theory of phase transitions and explain how to implement Markov chain Monte Carlo simulations and perform them for different lattice sizes with periodic boundary conditions. Energy, magnetization, specific heat, magnetic susceptibility and the correlation function are calculated and the critical exponents determined by finite-size scaling techniques. The importance of the correlation length as the relevant parameter in phase transitions is emphasized.
Today, the Ising model is an archetype describing collective ordering processes. As such, it is widely known in physics and far beyond. Less known is the fact that the thesis defended by Ernst Ising 100 years ago (in 1924) contained not only the solution of what we call now the ‘classical 1D Ising model’ but also other problems. Some of these problems, as well as the method of their solution, are the subject of this note. In particular, we discuss the combinatorial method Ernst Ising used to calculate the partition function for a chain of elementary magnets. In the thermodynamic limit, this method leads to the result that the partition function is given by the roots of a certain polynomial. We explicitly show that ‘Ising’s roots’ that arise within the combinatorial treatment are also recovered by the eigenvalues of the transfer matrix, a concept that was introduced much later. Moreover, we discuss the generalization of the two-state model to a three-state one presented in Ising’s thesis, which is not included in his famous paper of 1925 (E. Ising, Z. Physik 31 (1925) 253). The latter model can be considered as a forerunner of the now-abundant models with many-component order parameters.
Abstract Having been linked to numerous applications, the one-dimensional Lenz–Ising model in a non-zero magnetic field is well-known in graduate- and senior-level statistical physics. The transfer matrix method is the standard way to solve the model. In this line, we reformulate the transfer matrix by subjecting the usual Pauli matrices to rotation. The transfer matrix is then shown as a ‘spin’-representative of a two-dimensional rotation just like the special case when the magnetic field is zero. Consequently, eigenvalues in exponential form are obtained, and the zeroes of the partition function lie on a unit circle in agreement with the Lee–Yang Circle Theorem. The zeroes pinching the positive side of the real axis give the known trivial transition temperature, T = 0 K. Finally, we discuss the significance of the results in teaching the model.
Using a method suggested by Montroll, we extend the well-known matrix formulation of the nearest-neighbor one-dimensional Ising problem to allow for interactions with an arbitrary finite range n, general spin l, and an applied magnetic field B. We exhibit the relevant matrix element explicitly and hence formally obtain the partition function via an eigenvalue problem of order (2l + 1)n. For the case B = 0, l = ½ we introduce a change of variable which simplifies the partition function while still allowing a matrix formulation. Using this approach we have computed specific-heat curves for infinite, ferromagnetic Ising chains with interactions of range n (n ≤ 7). We prove in an appendix that open and cyclic boundary conditions are equivalent for the system under consideration.